Modelling and pricing derivatives on precipitation

The modelling and analysis of precipitation derivatives are based on different stochastic models than the CARMA processes used for wind speed and temperature. The modelling of precipitation involves the application of independent increment processes. Independent increment processes are an extension of the class of Lévy processes, to which Brownian motion belongs. We construct a continuous-time analogue of the time series model proposed in Sect. 3.3, and apply it for pricing various derivatives contracts. The market price of risk will be introduced via the Esscher transform, which preserves the independent increment property of the driving processes. Furthermore, we derive semi-analytical pricing formulas for derivatives like swaps and options based on Fourier transform methods and properties of independent increment processes. This chapter provides the basic background required on independent increment processes. For those not familiar with these processes and wishing to learn more, we refer to [Benth, Šaltytė Benth and Koekebakker (2008)] and the references therein for an introduction in the context of energy markets. See also [Cont and Tankov (2004)] for the theory on the special case of Lévy processes and their applications to finance, and [Jacod and Shiryaev (1987)] for a complete account on independent increment processes.